60 research outputs found
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Many PDEs involving fractional Laplacian are naturally set in unbounded
domains with underlying solutions decay very slowly, subject to certain power
laws. Their numerical solutions are under-explored. This paper aims at
developing accurate spectral methods using rational basis (or modified mapped
Gegenbauer functions) for such models in unbounded domains. The main building
block of the spectral algorithms is the explicit representations for the
Fourier transform and fractional Laplacian of the rational basis, derived from
some useful integral identites related to modified Bessel functions. With these
at our disposal, we can construct rational spectral-Galerkin and direct
collocation schemes by pre-computing the associated fractional differentiation
matrices. We obtain optimal error estimates of rational spectral approximation
in the fractional Sobolev spaces, and analyze the optimal convergence of the
proposed Galerkin scheme. We also provide ample numerical results to show that
the rational method outperforms the Hermite function approach
Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains
In this paper, we propose a fast spectral-Galerkin method for solving PDEs
involving integral fractional Laplacian in , which is built upon
two essential components: (i) the Dunford-Taylor formulation of the fractional
Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions
(MCFs) as basis functions. As a result, the fractional Laplacian can be fully
diagonalised, and the complexity of solving an elliptic fractional PDE is
quasi-optimal, i.e., with being the number of modes in
each spatial direction. Ample numerical tests for various decaying exact
solutions show that the convergence of the fast solver perfectly matches the
order of theoretical error estimates. With a suitable time-discretization, the
fast solver can be directly applied to a large class of nonlinear fractional
PDEs. As an example, we solve the fractional nonlinear Schr{\"o}dinger equation
by using the fourth-order time-splitting method together with the proposed
MCF-spectral-Galerkin method.Comment: This article has a total of 24 pages and including 22 figure
Performance of affine-splitting pseudo-spectral methods for fractional complex Ginzburg-Landau equations
In this paper, we evaluate the performance of novel numerical methods for
solving one-dimensional nonlinear fractional dispersive and dissipative
evolution equations. The methods are based on affine combinations of
time-splitting integrators and pseudo-spectral discretizations using Hermite
and Fourier expansions. We show the effectiveness of the proposed methods by
numerically computing the dynamics of soliton solutions of the the standard and
fractional variants of the nonlinear Schr\"odinger equation (NLSE) and the
complex Ginzburg-Landau equation (CGLE), and by comparing the results with
those obtained by standard splitting integrators. An exhaustive numerical
investigation shows that the new technique is competitive with traditional
composition-splitting schemes for the case of Hamiltonian problems both in
terms accuracy and computational cost. Moreover, it is applicable
straightforwardly to irreversible models, outperforming high-order symplectic
integrators which could become unstable due to their need of negative time
steps. Finally, we discuss potential improvements of the numerical methods
aimed to increase their efficiency, and possible applications to the
investigation of dissipative solitons that arise in nonlinear optical systems
of contemporary interest. Overall, our method offers a promising alternative
for solving a wide range of evolutionary partial differential equations.Comment: 31 pages, 12 figure
Paley-Littlewood decomposition for sectorial operators and interpolation spaces
We prove Paley-Littlewood decompositions for the scales of fractional powers
of -sectorial operators on a Banach space which correspond to
Triebel-Lizorkin spaces and the scale of Besov spaces if is the classical
Laplace operator on We use the -calculus,
spectral multiplier theorems and generalized square functions on Banach spaces
and apply our results to Laplace-type operators on manifolds and graphs,
Schr\"odinger operators and Hermite expansion.We also give variants of these
results for bisectorial operators and for generators of groups with a bounded
-calculus on strips.Comment: 2nd version to appear in Mathematische Nachrichten, Mathematical News
/ Mathematische Nachrichten, Wiley-VCH Verlag, 201
Spectral methods for nonlocal wave problems.
Masters Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF
New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus
This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention
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